Dandruff is different than dry scalp. Like SD, it is caused by an overgrowth of a normal yeast (Malassezia yeast) that lives on your scalp so it won't go away with co-washing. Co-washing may even make it worse. You'll have to use medicated shampoos.
The best thing to do in your case is to visit a derm, who will adequately assess your case and prescribe a solution.
The info to follow is simply for information. I am not a dermatologist and cannot assess your individual situation. The info is also not a replacement for a professional opinion, which I highly recommend you do seek.
Some have had success with tea tree oil. Personally I've used a shampoo with it (Giovanni TTTT) and while I liked the shampoo fine it didn't cure my dandruff. I also have occasional bouts of SD which I treat with Selsun Blue. I use it 1-3 times and it goes away. Sometimes it will come back 4 months later, sometimes it won't come back for 8 months. Selsun Blue is completely not CG though, but luckily I only have to use it a couple of times a year.
A derm recently told me to use in rotation shampoos containing:
Head & Shoulders (dark blue bottle)
Head & Shoulders (white bottle)
Dandruff and SD are recurring conditions so unfortunately you will have to keep using medicated shampoos, at the very least in regular intervals. You will most likely not be able to go completely CG. One thing you can do is coat up the lengths of your hair in conditioner and then concentrate the medicated shampoos on your scalp. When you rinse your scalp, the conditioner on your lengths should protect it from the sulfates usually found in the medicated poos.
I really do wish they would just freaking make a Head & Shoulders without the sulfates!!!!! I think I will write to them. The thing is, for me at least, harsh shampoos really aggravate my dandruff/SD. I think it's because it sends my scalp into producing more oil which then feeds the yeast.
There is evidence of using raw honey to treat SD. I don't know how it fares with dandruff but here's the link to the study for perusing.
Therapeutic and prophylactic effects of crude ... [Eur J Med Res. 2001] - PubMed - NCBI
I hope this is helpful! I really can't stress enough how important it is to visit your dermatologist for a proper diagnosis on any scalp issue, whatever it may be. Please go visit your derm!